On chromatic number of graphs and set systems

AbstractWe explore a general method based on trees of elementary submodels in order to present highly simplified proofs to numerous results in infinite combinatorics. While countable elementary submodels have been employed in such settings already, we significantly broaden this framework by developing the corresponding technique for countably closed models of size continuum. The applications range from various theorems on paradoxical decompositions of the plane, to coloring sparse set systems, results on graph chromatic number and constructions from point-set topology. Our main purpose is to demonstrate the ease and wide applicability of this method in a form accessible to anyone with a basic background in set theory and logic.

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On chromatic number of graphs and set-systems

Acta Mathematica Academiae Scientiarum Hungaricae ◽

10.1007/bf02020444 ◽

1966 ◽

Vol 17 (1-2) ◽

pp. 61-99 ◽

Cited By ~ 247

Author(s):

P. Erdős ◽

A. Hajnal

Keyword(s):

Chromatic Number ◽

Set Systems

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On chromatic number of finite set-systems

Acta Mathematica Academiae Scientiarum Hungaricae ◽

10.1007/bf01894680 ◽

1968 ◽

Vol 19 (1-2) ◽

pp. 59-67 ◽

Cited By ~ 103

Author(s):

L. Lovász

Keyword(s):

Chromatic Number ◽

Set Systems ◽

Finite Set

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Set Systems with Finite Chromatic Number

European Journal of Combinatorics ◽

10.1016/s0195-6698(89)80071-x ◽

1989 ◽

Vol 10 (6) ◽

pp. 543-549 ◽

Cited By ~ 3

Author(s):

Péter Komjáth

Keyword(s):

Chromatic Number ◽

Set Systems

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On the chromatic number of set systems

Random Structures and Algorithms ◽

10.1002/rsa.1020 ◽

2001 ◽

Vol 19 (2) ◽

pp. 87-98 ◽

Cited By ~ 20

Author(s):

Alexandr Kostochka ◽

Dhruv Mubayi ◽

Vojtěch Rödl ◽

Prasad Tetali

Keyword(s):

Chromatic Number ◽

Set Systems

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On A Combinatorial Problem III

Canadian Mathematical Bulletin ◽

10.4153/cmb-1969-051-5 ◽

1969 ◽

Vol 12 (4) ◽

pp. 413-416 ◽

Cited By ~ 17

Author(s):

P. Erdős

Keyword(s):

Chromatic Number ◽

Combinatorial Problem ◽

Acta Math ◽

Set Systems ◽

Property B

A family of sets {Aα} is said by Miller [3] to have property B if there exists a set S which meets all the sets Aα and contains none of them. Property B has been extensively studied in several recent papers (see the references in [2] and the last chapter of P. Erdös and A. Hajnal, On chromatic number of graphs and set systems, Acta. Math. Acad. Sci. Hung. 17 (1966) 61–99).

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Chromatic Numbers of Stable Kneser Hypergraphs via Topological Tverberg-Type Theorems

International Mathematics Research Notices ◽

10.1093/imrn/rny135 ◽

2018 ◽

Vol 2020 (13) ◽

pp. 4037-4061 ◽

Cited By ~ 2

Author(s):

Florian Frick

Keyword(s):

Euclidean Space ◽

Chromatic Number ◽

Lower Bounds ◽

Pairwise Disjoint ◽

Convex Hulls ◽

Chromatic Numbers ◽

Set Systems ◽

Equivariant Topology ◽

Disjoint Sets ◽

Combination Of Methods

Abstract Kneser’s 1955 conjecture—proven by Lovász in 1978—asserts that in any partition of the $k$-subsets of $\{1, 2, \dots , n\}$ into $n-2k+1$ parts, one part contains two disjoint sets. Schrijver showed that one can restrict to significantly fewer $k$-sets and still observe the same intersection pattern. Alon, Frankl, and Lovász proved a different generalization of Kneser’s conjecture for $r$ pairwise disjoint sets. Dolnikov generalized Lovász’ result to arbitrary set systems, while Kříž did the same for the $r$-fold extension of Kneser’s conjecture. Here we prove a common generalization of all of these results. Moreover, we prove additional strengthenings by determining the chromatic number of certain sparse stable Kneser hypergraphs, and further develop a general approach to establishing lower bounds for chromatic numbers of hypergraphs using a combination of methods from equivariant topology and intersection results for convex hulls of points in Euclidean space.

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